Is it legitimate to treat matrices this way?

[pivoxa15]

Make a matrix G so that each element in G is another matrix with a constant dimension (i.e. 3 by 3). Hence a 3 by 3 G would mean each element in G is a 3 by 3 matrix. With a total of 9 matrices in G.

Or a 6 by 6 G would mean each element in G for which there are 36 are 36 matrices each of which is 3 by 3.

If so than how about making each element in a matrix, vectors with a fixed dimension? So a 6 by 6 matrix would have 36 vectors with a fixed dimension.

Then matrices like G can be operated on just like scalar matrices but care has to be taken when considering the dimensions of the matrix elements in G. But when all this has been taken into account than would it all be okay?

 

[pivoxa15]

No reply in two days? Do I need to clarify or elaborate on my question?

Wiki has 'In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied.'

http://en.wikipedia.org/wiki/Matrix_(mathematics)

A matrix is itself abstract and forms a Ring? (since the hyperlink to 'abstract quantities that can be added and multiplied' is the Ring article).

 

[Galileo]

If you have a ring, then you can form the ring $M_n(R)$ of nxn matrices with entries in R. So you can make matrix rings of matrix rings etc. Is that what you meant?

 

[pivoxa15]

I was thinking something like $G_a(M_b(R))$. If the Latex is not showing click the formula to view. G and M denote square matrices. While a and b represent their dimensions respectively (i.e. axa bxb). Each element in G is matrix M with a dimension bxb. In M each element is in R.

 

[matt grime]

Of course you can do that.

And "a matrix" is not a ring. A matrix is a matrix. A collection of matrices can form a ring, and as Galileo says, the full collection of all matrices over some ring is again a ring.

 

[pivoxa15]

Thanks for the confirmation. I used this in an assigment but only realized whether it was legitimate after I handed it in. Because at the time, everything fitted so nicely that I didn't bother to think about the legitamcy of it.

 

[HallsofIvy]

Sounds like you are talking about "block matrices". It is not at all uncommon to take, for example, a 16 by 16 matrix and treat it as a 4 by 4 matrix, each of whose entries is a 4 by 4 matrix. It is also often done to treat an (n+1) by (n+1) matrix as a 2 by 2 matrix
$$\left[\begin{array}{cc}A & B \\ C & D\end{array}\right]$$
where A is an n by n matrix, B is an n by 1 matrix, C is a 1 by n matrix, and D is a 1 by 1 matrix.

In particular, one can show that the usual multiplication of matrices:
$$\left[\begin{array}{cc}A & B \\C & D\end{array}\right]\left[\begin{array}{cc}E & F \\G & H\end{array}\right] = \left[\begin{array}{cc}AE+ BG & AF+ BH \\ CE+ CG & CF+ DH\end{array}\right]$$
is still valid as long as the "sub matrices" are of the correct sizes to multiply..

 

[pivoxa15]

On the subject of abstract entities, what are the major differences between a ring and a group? A ring has more axioms than a group but the axioms are of very similar in nature (operations in a ring is restricted to addition and multiplication). But the operation in a group is unrestricted. What are the relations between groups and rings? Is a group a more general entity than a ring hence rings are also groups but the opposite does not occur often?

Also are fields normally a subset of a ring?

 

[loopgrav]

Get a copy of Hungerford's Abstract Algebra.

 

[pivoxa15]

Get a copy of Hungerford's Abstract Algebra.

Incidently, I have a copy right in my home because I am doing an intro subject on this stuff. Is there a reason why you recommand this one over other intro algebra books?

We just started on abstract algebra and covered a little bit on groups and fields but not rings yet. I just like some general answers (if possible) for my previous question about groups and rings.

 

[matt grime]

On the subject of abstract entities, what are the major differences between a ring and a group? A ring has more axioms than a group but the axioms are of very similar in nature (operations in a ring is restricted to addition and multiplication).

since you have the axioms in front of you, you know the differences.

But the operation in a group is unrestricted.

I have no idea what that means.

What are the relations between groups and rings?

A ring is a set with two binary operations, like the integers, or the rationals, or matrices. A group is a set with one operation, like the complex nth roots of unity under multiplication, or the set of bijections from a set to itself (composition).

Is a group a more general entity than a ring hence rings are also groups but the opposite does not occur often?

I'm not sure it makes sense to think about things like that.

Also are fields normally a subset of a ring?

A field is a ring. I don't know what you mean by that question again.